Question: Simplify and expand the following expression: $ \dfrac{1}{2z + 4}+\dfrac{2z - 7}{5z - 1} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2z + 4)(5z - 1)$ Multiply the first term by $\dfrac{5z - 1}{5z - 1}$ $ \begin{align*} \dfrac{1}{2z + 4} \times \dfrac{5z - 1}{5z - 1} & = \dfrac{(1)(5z - 1)}{(2z + 4)(5z - 1)} \\ & = \dfrac{5z - 1}{(2z + 4)(5z - 1)}\end{align*} $ Multiply the second term by $\dfrac{2z + 4}{2z + 4}$ $ \begin{align*} \dfrac{2z - 7}{5z - 1} \times \dfrac{2z + 4}{2z + 4} & = \dfrac{(2z - 7)(2z + 4)}{(5z - 1)(2z + 4)} \\ & = \dfrac{4z^2 - 6z - 28}{(5z - 1)(2z + 4)}\end{align*} $ Now we have: $ = \dfrac{5z - 1}{(2z + 4)(5z - 1)} + \dfrac{4z^2 - 6z - 28}{(5z - 1)(2z + 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5z - 1 + 4z^2 - 6z - 28}{(2z + 4)(5z - 1)} $ $ = \dfrac{-z - 29 + 4z^2}{(2z + 4)(5z - 1)}$ Expand the denominator: $ = \dfrac{-z - 29 + 4z^2}{10z^2 + 18z - 4}$